For the stock keyboard, go into Settings -> Language & input and select the settings icon next to Android keyboard. Select Input languages at the top of the screen. Uncheck “Use system language”. In the Active Input Methods section below, leave English (US) checked, then scroll down and put a check next to Russian.

After that, a Globe button shows up next to the spacebar. Pressing it toggles between all of your selected languages

In case you aren’t aware, this also works with the Google voice typing! To do this, select the “Choose input languages” option in the Google voice typing settings, then pick your languages.